!Lecture #7& 8.

 

Definition 1. The conditional probability of an event A, given that an event B has occurred, is equal to

P(A | B) = P(A∩B)/ P(B)

 

provided P(B) > 0.

 

Definition 2. Two events, A and B, are said to be independent if

 

P(A∩B)= P(A) P(B)

 

For nonnull events, A and B, the definition 2 is equivalent to

 

P(A | B)= P(A)  or P(B | A)= P(B)

 

Multiplicative  Law of Probability

 

P(A∩B) = P(A) P(B | A) = P(B) P(A | B)

 

 

The Law of Total Probability

If  A1, A2,A3,  … form a sequence of pairwise mutually exclusive and exhaustive events in S (Ai ∩ Aj = Ø , i ≠j ) then

 

P(B) = ∑ P(B|Ai) P(Ai)

                                 i

 

BAYES’ THEOREM

 

 

P(Aj |B) = [P(B|Aj) P(Aj)]/∑ P(B|Ai) P(Ai)                                                                                                                             

                                                                     i

 

 

j = 1,2,…,k    with P(Aj) > 0 and P(B) > 0

 

 

 

 

 

 

 

SHORTCUT IF A PROBLEM INVOLVES

TWO EVENTS   A   and   B

 

A

A′

 

 

B

P(A ∩ B)

P(A′∩B)

 

P(B)

B′

P(A ∩ B′)

P(A′∩ B′ )

 

P(B′)

 

P(A)

P(A′)

1

 

 

 

 

 

Note:

P(A) + P(A′) = 1                             P(B) + P(B′) = 1

P(A ∩ B) + P(A′∩B) = P(B)          P(A ∩ B′) + P(A′∩ B′ ) = P(B′)

P(A ∩ B) + P(A ∩ B′) = P(A)        P(A′∩B) + P(A′∩ B′ ) =P(A′)

 

The event-composition approach to calculate the probability of an event of interest, say event A, expresses A as a composition (unions and/or intersections) of two or more other events. Then the two laws of probability can be applied to the composition to find P(A). Mutually exclusive events simplify the addition law and independence simplifies the multiplication law of probability.

The best way to learn how to solve probability problems is to learn by doing!

 

 

PROBABILITY OF A COMPOSITION OF TWO OR MORE EVENTS.   EXAMPLES.

 

Example1. In how many ways can 7 students line up outside Statistics Professor’s door to complain about grades?

 

Example 2. A box contains 4 red balls and 6 other balls. Two balls are drawn

without replacement. What is the probability that they are red?

 

Example 3. A box contains 8 red balls and 14 black balls. Three balls are drawn

without replacement. (a) What is the probability that all 3  are red?

(b) What is the probability that 1 is  red and 2 are black?

 

Example 4. In a certain community 30% of the people smoke, 55% of them drink alcohol, and 20% of them smoke as well as drink. Calculate the probability that a randomly selected person

(i)                  smokes but does not drink

(ii)                neither smokes nor drink

(iii)               either smokes or does not drink or both

 

 

 

 

Example 5 . A bank has five junior executive in the head branch. Each year, one of these five is selected at random to be transferred  as manager to one of the local branches. The selected individual is replaced by a new junior executive. Find the probability that a given junior executive:

(i)                  Stays exactly 2 years in the head branch

(ii)                Stays more than 2 years in the head branch

(iii)               Stays 3 years or less in the head branch.

 

Example 6 . In a large population of fruit flies, 30% of the flies have a wing mutation, 40% have an eye mutation, and 15% have both eye and wing mutations What is the probability that a fly has at least one of the mutations?

 

Example 7c .  A basket contains 2 red balls and 2 white balls. A ball is drawn and set aside, and its color is noted. Then a second ball is drawn. What is the probability that both are red?

 

Example 8ci. A card is drawn from a deck of cards. Then the card is replaced, the deck is reshuffled, and a second card is drawn. What is the probability of  getting an ace on the first and a king on the second?

 

Example 9c.The experiment involves the selection of  2 applicants out of 5. Find the

probability of drawing exactly one of the two best applicants, event A.

 

Example 10c. A monkey is to be taught to recognize colors by tossing one red, one

black, and one white ball into boxes of the same respective colors, one ball to a box.

If the monkey has not learned the colors, and merely tosses one ball into each box at

random, find the probability of  (a) No color matches. (b)Exactly one color match.

 

Example 11b. In answering a question on a multiple choice test a student either knows the answer or he guesses. Let p be the probability that he knows the answer. Assume that a student who guesses at the answer will be correct with probability  1/m , where m is the number of multiple choice alternatives. What is the conditional probability that a student knew the answer to a question given that he answered it correctly?

 

Example 12b. An oil company estimates that from geological data there is a probability of 0.3 of finding oil in a certain area. It knows from previous experience that if oil is to be found, there is a probability of 0.4 that a positive strike of some kind will be made on the first series of drillings. If the first series of drillings turn out to be negative, what is the probability that oil will eventually be found? Suppose that a second drillings turn out to be negative. What then will be the new probability of eventually striking oil, if (a) assuming that it knows from previous experience that  every series of strikes is independent; (b) assuming that it knows from previous experience that if oil is to be found, there is a probability of 0.6 that a positive strike of some kind will be made on the second series of drillings?

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